PNNL

Abstract

We describe quantum circuits with only O(N) Toffoli complexity that block encode the spectraof quantum chemistry Hamiltonians in a basis ofNmolecular orbitals. With O(?/epsilon) repetitions of these circuits one can use phase estimation to sample in the molecular eigenbasis, where?is the1-norm of Hamiltonian coefficients and epsilon is the target precision. This is the lowest complexity that has been shown for quantum computations of chemistry within an arbitrary basis. Furthermore, up to logarithmic factors, this matches the scaling of the most efficient prior block encodings that can only work with orthogonal basis functions diagonalizing the Coloumb operator (e.g., the planewave dual basis). Our key insight is to factorize the Hamiltonian using a method known as tensor hypercontraction (THC) and then to transform the Coulomb operator into a diagonal form witha non-orthogonal basis defined by THC factors. We then use qubitization to simulate the non-orthogonal THC Hamiltonian, in a fashion that avoids most complications of the non-orthogonalbasis. We also reanalyze and reduce the cost of several of the best prior algorithms for these simulations in order to facilitate a clear comparison to the present work. In addition to having lower asymptotic scaling spacetime volume, compilation of our algorithm for challenging finite-sized molecules such as FeMoCo reveals that our method requires the least fault-tolerant resources of any known approach. By laying out and optimizing the surface code resources required of our approach we show that FeMoCo can be simulated using about four million physical qubits and under four days of runtime, assuming 1µs cycle times and physical gate error rates no worse than 0.1%.